Statistical properties of the two dimensional Feshbach–Villars oscillator (FVO) in the Rotating Cosmic String space-time

In this section, we deal with the relativistic quantum description of spin-0 particle propagating in Minkowski space-time with the metric tensor $\eta_{\mu\nu}=\text{diag}\left(1,-1,-1,-1\right)$. For a scalar massive particle $\Phi$ with mass $m>0$, the standard covariant KG equation reads (key-41, ; key-42, )

$$\left(\eta^{\mu\nu}D_{\mu}D_{\nu}+m^{2}\right)\Phi(x,t)=0,$$

(1)

where $D_{\mu}=i\left(p_{\mu}-eA_{\mu}\right)$ denotes the minimally coupled covariant derivative. $p_{\mu}=\left(E,-p_{i}\right)$ is the canonical four momentum and $A_{\mu}=\left(A_{0},-A_{i}\right)$ is the electromagnetic four potential, respectively. $e$ is the magnitude of the particle charge.

At this place, it is worth emphasizing that Eq. ((1)) can be rewritten in a Hamiltonian form with the time first derivative i.e, a Schrodinger’s equation

$$\mathcal{H}\Phi\left(x,t\right)=i\,\frac{\partial}{\partial t}\Phi(x,t).$$

(2)

The Hamiltonian $\mathcal{H}$ can be defined by the use of the FV linearization procedure i.e, transforming Eq. (1) to a first order in time differential equation. We introduce the two component wave function (key-43, ; key-44, ),

$$\Phi(x,t)=\left(\begin{array}[]{c}\phi_{1}(x,t)\\ \phi_{2}(x,t)\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1+\frac{i}{m}\mathcal{D}\\ 1-\frac{i}{m}\mathcal{D}\end{array}\right)\psi(x,t),$$

(3)

here $\psi(x,t)$ obeys the KG wave equation, and we have defined $\mathcal{D}$ such that

$$\mathcal{D}=\frac{\partial}{\partial t}+ieA_{0}(x),$$

(4)

the above transformation ((3)) consists in introducing wave functions satisfying the conditions

$$\psi=\phi_{1}+\phi_{2},\qquad i\mathcal{D}\psi=m\left(\phi_{1}-\phi_{2}\right).$$

(5)

For our subsequent review, it is convenient to write

	$\displaystyle\phi_{1}$	$\displaystyle=\frac{1}{2m}\left[m+i\frac{\partial}{\partial t}-eA_{0}\right]\psi$		(6)
	$\displaystyle\phi_{2}$	$\displaystyle=\frac{1}{2m}\left[m-i\frac{\partial}{\partial t}+eA_{0}\right]\psi,$

Equivalently, Eq. ((1)) becomes

	$\displaystyle\left[i\frac{\partial}{\partial t}-eA_{0}\right]\left(\phi_{1}+\phi_{2}\right)$	$\displaystyle=m\left(\phi_{1}-\phi_{2}\right)$		(7)
	$\displaystyle\left[i\frac{\partial}{\partial t}-eA_{0}\right]\left(\phi_{1}-\phi_{2}\right)$	$\displaystyle=\left[\frac{\left(p_{i}-eA_{i}\right)^{2}}{m}+m\right]\left(\phi_{1}+\phi_{2}\right),$

Addition and subtraction of these two equations lead to the system of coupled differential equations of first order in time,

	$\displaystyle\frac{\left(p_{i}-eA_{i}\right)^{2}}{2m}\left(\phi_{1}+\phi_{2}\right)+\left(m+eA_{0}\right)\phi_{1}$	$\displaystyle=i\frac{\partial\phi_{1}}{\partial t}$		(8)
	$\displaystyle\frac{-\left(p_{i}-eA_{i}\right)^{2}}{2m}\left(\phi_{1}+\phi_{2}\right)-\left(m-eA_{0}\right)\phi_{2}$	$\displaystyle=i\frac{\partial\phi_{2}}{\partial t},$

Using Eqs. ((8)), the FV Hamiltonian of a scalar particle in the presence of the electromagnetic interaction may be written as

$$\mathcal{H}_{KG}=\left(\tau_{z}+i\tau_{y}\right)\frac{\left(p_{i}-eA_{i}\right)^{2}}{2m}+m\tau_{z}+eA_{0}(x),$$

(9)

where $\tau_{i}\,\left(i=x,y,z\right)$ are the conventional $2\times 2$ Pauli matrices given by

$$\tau_{x}=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right),\quad\tau_{y}=\left(\begin{array}[]{cc}0&-i\\ i&0\end{array}\right),\quad\tau_{z}=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right).$$

(10)

It is noteworthy that the Hamiltonian ((9)) fulfills the generalized hermicity condition ¹¹1The Hamiltonian $\mathcal{H}$ is said to be pseudo-Hermitian if there is an invertible, Hermitian, linear operator $\beta$ such that $\mathcal{H}^{\dagger}=\beta\mathcal{H}\beta^{-1}$ (key-45, ),

$$\mathcal{H}_{KG}=\tau_{z}\mathcal{H}_{KG}^{\dagger}\tau_{z},\qquad\mathcal{H}_{KG}^{\dagger}=\tau_{z}\mathcal{H}_{KG}\tau_{z}.$$

(11)

For the free particle propagation i.e, no interaction is considered $\left(A_{\mu}=0\right),$ the one dimensional FV Hamiltonian simplifies to

$$\mathcal{H}_{0}=\left(\tau_{z}+i\tau_{y}\right)\frac{p_{x}^{2}}{2m}+m\tau_{z},$$

(12)

The solutions to the free Hamiltonian (which is independent of time) are just stationary states. Assuming a solution of the form (key-36, ),

$$\Phi\left(x,t\right)=\Phi\left(x\right)e^{-iEt}=\left(\begin{array}[]{c}\phi_{1}\left(x\right)\\ \phi_{2}\left(x\right)\end{array}\right)e^{-iEt},$$

(13)

with $E$ being the energy of the system. Thus, Eq. ((2)) can then be written as

$$\mathcal{H}_{0}\Phi\left(x\right)=E\Phi\left(x\right),$$

(14)

which is the one-dimensional FV equation of the free relativistic spin-0 particle and it is carried out with the aim of having an alternative Schrodinger’s to KG equation. In what follows, the above method will be used to find the solutions of wave equations in curved space-time, namely, the cosmic string.

For the next discussion, it is preferable to analyze the KGO in Minkowski space-time exploiting the method described previously. In order to examine the KGO in the FV representation, we start with the minimal substitution $p_{\mu}\longrightarrow p_{\mu}+im\omega x_{\mu}$ of the momentum operator (key-46, ). Here $\omega$ is the oscillation frequency and $x_{\mu}=(0,x,0,0)$ . Thus, by generalizing the momentum operator, the one dimensional FV Hamiltonian ((12)) becomes

$$\mathcal{H}_{KGO}=\frac{1}{2m}\left(\tau_{z}+i\tau_{y}\right)\left(P_{x}-im\omega x\right)\left(P_{x}+im\omega x\right)+m\tau_{z},$$

(15)

In this way, one obtains the Schrödinger formulation of KGO in one dimension by substituting the Ansatz ((13)) into Eq. ((14)) by means of Eq. ((15)),

	$\displaystyle\frac{1}{2m}\left[\frac{d^{2}}{dx^{2}}-m^{2}\omega^{2}x^{2}+m\omega\right]\left(\phi_{1}+\phi_{2}\right)+m\phi_{1}$	$\displaystyle=E\phi_{1}$		(16)
	$\displaystyle-\frac{1}{2m}\left[\frac{d^{2}}{dx^{2}}-m^{2}\omega^{2}x^{2}+m\omega\right]\left(\phi_{1}+\phi_{2}\right)-m\phi_{2}$	$\displaystyle=E\phi_{2},$

after performing a calculation similar to the one was done in the previous KG equation case, then it follows that

$$\left[\frac{d^{2}}{dx^{2}}-m^{2}\omega^{2}x^{2}+\left(E^{2}-m^{2}+m\omega\right)\right]\psi\left(x\right)=0,$$

(17)

or,

$$\left[\frac{d^{2}}{dx^{2}}-\lambda_{1}x^{2}+\lambda_{2}\right]\psi\left(x\right)=0$$

(18)

where we have set,

$$\lambda_{1}=m^{2}\omega^{2},\qquad\lambda_{2}=E^{2}-m^{2}+m\omega.$$

(19)

Eq. ((18)) is a second order differential equation for the field $\psi$ describing the KGO dynamics in one dimensional Minkowski space-time. The solution of the above equation can be found in literature, and a quantization condition of the energy is followed from that solution, providing

$$\frac{E^{2}-m^{2}+m\omega}{2m}=\left(n+\frac{1}{2}\right)\omega,\qquad n=0,1,2,\cdots.$$

(20)

After arranging and simplifying the condition ((20)), we find the following expression of the energy spectrum

$$E=\pm\sqrt{2m\omega n+m^{2}},$$

(21)

Eq.((21)) presents the relativistic energy spectrum of KGO in Minkowski space-time.

Before we study the KGO in the Hamiltonian representation, let us first derive the KG wave equation for the free relativistic scalar particle propagating in the cosmic string space-time that is assumed to be static and cylindrical symmetric.

The general expression for a (3+1)-dimensional cosmic string metric is defined by the line element (key-47, ; key-48, )

	$\displaystyle ds^{2}$	$\displaystyle=g_{\mu\nu}dx^{\mu}dx^{\nu}$
		$\displaystyle=dt^{2}-dr^{2}-\alpha^{2}r^{2}d\varphi^{2}-dz,$		(24)

in cylindrical coordinates ²²2Note that this metric is an exact solution to Einstein’s field equations for $0\leq\mu<1/4,$ and by setting $\varphi^{\prime}=\alpha\varphi$, then it represents a flat conical exterior space with angle deficit $\delta\phi=8\pi\mu$.. Here $-\infty\leq t\leq+\infty$, $r\geq 0$, $0\leq\varphi\leq 2\pi$, $-\infty\leq z\leq+\infty,$ and $\alpha\in[0,1[\>$ is the angular parameter which determines the angular deficit $\delta\varphi=2\pi(1-\alpha)$, and it is related to the linear mass density $\mu$ of the string by $\alpha=1-4\mu$.

For the sake of simplicity in treating our quantum mechanical problem, let us work in a lower dimensional space in which $z=\text{cont}$, and because there is no structure in $z$-direction, we can suppress it (key-49, ; key-50, ). Thus, the metric of a static cosmic string with cylindrical symmetry has a $(2+1)$-dimensional form ³³3Since this metric is Lorentz-invariant under boosts in the $(t,z)$ plane (key-18, ; key-48, ), and by the virtue of rotational symmetry along the $z$-axis, it is reasonable to assume that the theory is invariant in the 2-dimentional $(r,\varphi)$ surface.

$$ds^{2}=dt^{2}-dr^{2}-\alpha^{2}r^{2}d\varphi^{2}$$

(25)

where the components of the metric and the inverse metric tensors are, respectively,

$$g_{\mu\nu}=\left(\begin{array}[]{ccc}1&0&0\\ 0&-1&0\\ 0&0&-\left(\alpha r\right)^{2}\end{array}\right),\quad g^{\mu\nu}=\left(\begin{array}[]{ccc}1&0&0\\ 0&-1&0\\ 0&0&\frac{-1}{\left(\alpha r\right)^{2}}\end{array}\right)$$

(26)

Its is worthy to mention that the subject of spinless massive particles in the geometry generated by a static cosmic string background has been discussed in several papers ( see, e.g, (key-25, ; key-51, ))

In what follows, we shall adopt the procedure presented in Refs. (key-52, ; key-53, ) to derive the FV form of KG wave equation in curved manifolds. We use the generalized Feshbach–Villars transformation (GFVT) ⁴⁴4An equivalent transformation was proposed earlier in Ref.(key-54, ) which is appropriate for describing both massive and massless particles. In the GFVT, the components of the wave function $\Phi$ are given by (key-52, )

$$\psi=\phi_{1}+\phi_{2},\qquad i\tilde{\mathcal{D}}\psi=\mathcal{N}\left(\phi_{1}-\phi_{2}\right),$$

(27)

where $\mathcal{N}$ is an arbitrary nonzero real parameter, and we have defined $\tilde{\mathcal{D}}=\frac{\partial}{\partial t}+\mathcal{Y},$ with

$$\mathcal{Y}=\frac{1}{2g^{00}\sqrt{-g}}\left\{\partial_{i},\sqrt{-g}g^{0i}\right\},$$

(28)

The curly bracket in Eq. ((28)) denotes the anti-commutator. For the above mentioned transformation, the Hamiltonian reads

$$\mathcal{H}_{GFVT}=\tau_{z}\left(\frac{\mathcal{N}^{2}+\mathcal{T}}{2\mathcal{N}}\right)+i\tau_{y}\left(\frac{-\mathcal{N}^{2}+\mathcal{T}}{2\mathcal{N}}\right)-i\mathcal{Y},$$

(29)

with

$$\mathcal{T}=\frac{1}{g^{00}\sqrt{-g}}\partial_{i}\sqrt{-g}g^{ij}\partial_{j}+\frac{m^{2}-\xi R}{g^{00}}-\mathcal{Y}^{2},$$

(30)

here and below $(i,j=1,2,3)$. We note that for $\mathcal{N}=m$, the original FV transformations are satisfied.

Now, considering the metric (25), it is easy to find that $R=0$, in other words, the space-time is locally flat (there is no local gravity), and hence the coupling term is vanishing⁵⁵5The case $\xi=0$ is refereed to as minimal coupling. However, for massless theory, $\xi$ takes the value 1/6 (in 4 dimensions). Then, in this later case, the equations of motion are conformally invariant..

A straightforward calculation leads to $\mathcal{Y}=0,$ then we obtain

$$\mathcal{T}=-\frac{d^{2}}{dr^{2}}-\frac{1}{r}\frac{d}{dr}-\frac{1}{\alpha^{2}r^{2}}\frac{d^{2}}{d\varphi^{2}}+m^{2},$$

(31)

Using these results to find the Hamiltonian (29), then because of the time and angular independence in the metric (25), one can assume a solution of the form ⁶⁶6we seek solutions that are cylindrical symmetric, i.e. solutions that have a rotational symmetry in the $(x,y)$-plane and do not explicitely depend on $z$.,

$$\Phi(t,r,\varphi)=\Phi(r)e^{-i\left(Et-j\varphi\right)},$$

(32)

where $j=0,\pm 1,\pm 2,..$ are the eigenvalues of the $z$component of the angular momentum operator. Therefore, it follows that the KG equation (22) may be written equivalently to the following two coupled equations

	$\displaystyle\left(\mathcal{N}^{2}+\mathcal{T}\right)\phi_{1}+\left(-\mathcal{N}^{2}+\mathcal{T}\right)\phi_{2}$	$\displaystyle=2\mathcal{N}E\phi_{1}$
	$\displaystyle-\left(\mathcal{N}^{2}+\mathcal{T}\right)\phi_{2}-\left(-\mathcal{N}^{2}+\mathcal{T}\right)\phi_{1}$	$\displaystyle=2\mathcal{N}E\phi_{2},$		(33)

The sum and the difference of the two last equations give a second order differential equation for the field $\psi$. Thus, the radial equation is written as follows

$$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{\zeta^{2}}{r^{2}}+\kappa\right]\psi\left(r\right)=0,$$

(34)

where we have set

$$\zeta=\frac{j}{\alpha},\qquad\kappa=\sqrt{E^{2}-m^{2}}.$$

(35)

We can observe that Eq. ((34)) is a Bessel equation and its general solution is defined by (key-55, )

$$\psi\left(r\right)=A\,J_{|\zeta|}\left(\kappa r\right)+B\,Y_{|\zeta|}\left(\kappa r\right),$$

(36)

where $J_{|\zeta|}\left(\kappa r\right)$ and $Y_{|\zeta|}\left(\kappa r\right)$ are the Bessel functions of order $\zeta$ and of the first and the second kind, respectively. Here $A$ and $B$ are arbitrary constants. We notice that at the origin when $\zeta=0$, the function $J_{|\zeta|}\left(\kappa r\right)\neq 0$. However, $Y_{|\zeta|}\left(\kappa r\right)$ is always divergent at the origin. In this case, we will consider only $J_{|\zeta|}\left(\kappa r\right)$ when $\zeta\neq 0$. Hence, we write the solution to Eq. ((34)) as follows

$$\psi\left(r\right)=A\,J_{\frac{|j|}{\alpha}}\left(\sqrt{E^{2}-m^{2}}\,r\right),$$

(37)

using this solution, we can now write the complete two-components wave-function of the spinless massive KG particle in the space-time of a static cosmic string

$$\psi\left(t,r,\varphi\right)=|C|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\ 1-\frac{E}{\mathcal{N}}\end{array}\right)e^{-i\left(Et-j\varphi\right)}\,J_{\frac{|j|}{\alpha}}\left(\sqrt{E^{2}-m^{2}}\,r\right),$$

(38)

The constant $|C|$ can be obtained by the appropriate normalization condition associated with the KG equation ( e.g, see Ref. (key-56, ; key-57, )), however, it is fortunate that non determining the normalization constants throughout this manuscript does not affect the final results.

We turn now to the particular case where we want to extend the GFVT for the KGO. In general, we need to perform a substitution of the momentum operator in Eq. ((22)). Consequently, it is possible to rewrite Eq. ((30)) as follows

$$\mathcal{T}=\frac{1}{\sqrt{-g}}\left(\frac{\partial}{\partial r}-m\omega r\right)\left(-\sqrt{-g}\right)\left(\frac{\partial}{\partial r}+m\omega r\right)-\frac{1}{\alpha^{2}r^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}+m^{2},$$

(39)

Similarly, a straightforward calculation based on the procedure that was carried out in the above discussion, one can obtain the following differential equation

$$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}+m^{2}\omega^{2}r^{2}-\frac{\sigma^{2}}{r^{2}}+\delta\right]\psi\left(r\right)=0,$$

(40)

with

$$\sigma^{2}=\left(\frac{j}{\alpha}\right)^{2},\qquad\delta=E^{2}-m^{2}+2m\omega.$$

(41)

Eq. ((40)) is the KGO for spin-0 particle in the space-time of a static cosmic string. To obtain the solution of this equation, we first propose a transformation of the radial coordinate

$$\chi=m\omega r^{2},$$

(42)

substituting the expression for $\chi$ into Eq. ((40)), we obtain

$$\left[\frac{d^{2}}{d\chi^{2}}+\frac{1}{\chi}\frac{\partial}{d\chi}-\frac{\sigma^{2}}{4\chi^{2}}+\frac{\delta}{4m\omega\chi}-\frac{1}{4}\right]\psi\left(\chi\right)=0.$$

(43)

Now, if we study the asymptotic behavior of the wave function at the origin and infinity, and since we are looking for regular solutions, we may suppose a solution that has the form

$$\psi\left(\chi\right)=\chi^{\frac{\left|\sigma\right|}{2}}e^{-\frac{\chi}{2}}F\left(\chi\right),$$

(44)

As before, this can be substituted back into Eq. ((43)), then we have

$$\chi\frac{d^{2}F\left(\chi\right)}{d\chi^{2}}+\left(|\sigma|+1-\chi\right)\frac{dF\left(\chi\right)}{d\chi}-\left(\frac{|\sigma|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2}\right)F\left(\chi\right)=0,$$

(45)

This is the confluent hyper-geometric equation (key-58, ) whose solutions are expressed in terms of the confluent hyper-geometric function type

$$F\left(\chi\right)=_{1}F_{1}\left(\frac{\left|\sigma\right|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2},+1,\chi\right),$$

(46)

We should note that the solution ((46)) must be a polynomial function of degree $n$. However, taking $n\rightarrow\infty$ imposes a divergence issue. We can have a finite polynomial only if the factor of the last term in Eq. ((45)) is a negative integer, meaning,

$$\frac{\left|\sigma\right|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2}=-n\qquad,n=0,1,2,\cdots.$$

(47)

Exploiting this result, and by the insertion of the parameters ((41)), we can obtain the quantified energy spectrum of KGO in the static cosmic string space-time, hence,

$$E\left(n,\alpha,j\right)=\pm\sqrt{4m\omega n+\frac{2m\omega\left|j\right|}{\alpha}+m^{2}},$$

(48)

We can observe that the energy depends explicitly on the angular deficit $\alpha$. In other words, the curvature of the space-time which is influenced by the topological defect i.e, the cosmic string through the deficit parameter will affects on the relativistic dynamics of the scalar particle by generating a gravitational field due to the presence of the wedge angle. Fig. (1) shows the energy levels of the KGO as a function of the quantum number $n$ in a static cosmic string space characterized by a wedge parameter $\alpha$ which takes different values.

The corresponding wave function is given by

$$\psi\left(t,r,\varphi\right)=|\tilde{C}|\left(m\omega r^{2}\right)^{\frac{\left|j\right|}{2\alpha}}e^{-\frac{m\omega r^{2}}{2}}{}_{1}F_{1}\left(\frac{\left|j\right|}{2\alpha}-\frac{\delta}{4m\omega}+\frac{1}{2},\frac{|j|}{\alpha}+1,m\omega r^{2}\right),$$

(49)

Then the general eigenfunctions are expressed as

$$\psi\left(t,r,\varphi\right)=|\tilde{C}|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\ 1-\frac{E}{\mathcal{N}}\end{array}\right)\left(m\omega r^{2}\right)^{\frac{\left|j\right|}{2\alpha}}e^{-\frac{m\omega r^{2}}{2}}e^{-i\left(Et-j\varphi\right)}{}_{1}F_{1}\left(\frac{\left|j\right|}{2\alpha}-\frac{\delta}{4m\omega}+\frac{1}{2},\frac{|j|}{\alpha}+1,m\omega r^{2}\right),$$

(50)

where $|\tilde{C}|$ is the normalization constant.

Having the complete wave-function above, we are in position to calculate the density $\rho_{{}_{KGO}}$ corresponds to the KGO given by (key-33, )

$$\rho_{{}_{KGO}}=\psi^{*}\tau_{z}\psi$$

(51)

In Fig. (2) we plot the density of KGO in static cosmic string as a function of the radial distance $r.$ Obviously, we can see that the density is affected by the choice of the variable $\mathcal{N}$.

In this section we shall analyze the KGO in the background geometry of a rotating cosmic string in three dimensions. Similarly to the case studied in Sec. III, the equations of motion of a scalar particle can be achieved by considering the GFVT. Several authors studied the quantum dynamics of relativistic particles in the space-time of a cosmic string with rotational effects, and many models were considered in this context. For instance, in a previous paper by Mazur (key-59, ), the quantum mechanical properties of massive (or massless) particles in the gravitational field of spinning cosmic strings were discussed. He showed that the energy should be quantized in the presence of non-zero angular momentum of the string. Later, Gerbert and Jackiw (key-60, ) have presented solutions for the KG and Dirac equations in the (2+1)-dimensional space-time created by a massive point particle, with arbitrary angular momentum. In Ref. (key-61, ), the vacuum expectation value of the stress-energy tensor for a massless scalar field conformally coupled to gravitation was discussed. The authors of Ref. (key-62, ) have examined the behavior of a quantum test particle satisfying the Klein-Gordon equation in a space-time of a spinning cosmic string. Additionally, it was shown in Ref. (key-63, ) that rotating cosmic string solution of $U(1)$ scalar field theory with a cylindrical symmetric energy density can be characterized as an extrema of the field’s energy for given angular and linear momenta. Moreover, topological and geometrical phases due to gravitational field of a cosmic string that has mass and angular momentum were investigated in Ref. (key-64, ).

Recently, the subject of gravitational effects of rotating cosmic strings has attracted considerable interest in connection with the dynamics of relativistic quantum particles and their properties. For example, vacuum fluctuations for a massless scalar field around a spinning cosmic string were studied in Ref. (key-65, ) applying a re-normalization method. In like manner, vacuum polarization of a scalar field in the gravitational background of a spinning cosmic string was investigated in Ref. (key-66, ). Furthermore, the authors of Ref. (key-67, ) have analyzed the Landau levels of a spinless massive particle in the spacetime of a rotating cosmic string by means of a fully relativistic approach. Wang et al. (key-68, ) have treated the model of the KGO coupled to a uniform magnetic field in the background of the rotating cosmic string. Also, the problem of a spinless relativistic particle subjected to a uniform magnetic field in the spinning cosmic string space-time was addressed in Ref.(key-69, ). Besides, the relativistic quantum dynamics of a KG scalar field subjected to a Cornell potential in spinning cosmic string space-time was presented in Ref. (key-70, ). In addition, the relativistic scalar charged particle in a rotating cosmic string space-time with Cornell-type potential and Aharonov-Bohm effect was analyzed in Ref. (key-29, ). Other publications in which rotating cosmic strings have been considered to study the quantum properties of relativistic systems could be mentioned in this respect (e.g, (key-71, ; key-72, ; key-73, ; key-74, ; key-75, ; key-76, ; key-77, ; key-78, ) and related references therein). Aside from investigating bosonic particles in that specific space-time, in the literature one can find various discussions concerning the effects of the topology and geometry of this space-time on the quantum dynamics of fermionic particles and their behaviour ( see for instance Refs. (key-79, ; key-80, ; key-81, ; key-82, ; key-83, ; key-84, ; key-85, )).

In the next discussion, we extend the problem studied in Sec. III to a more general space-time with non-zero angular momentum. We consider a massive, relativistic spin-0 particle whose wave-function is denoted by $\Psi$ and satisfies the KG equation 22 in the space-time induced by a (2+1)-dimensional stationary rotating cosmic string which is described by the following line element (key-86, ; key-87, ; key-59, ).

$$ds^{2}=dt^{2}+2a\,dt\,d\phi-dr^{2}-\left(\alpha^{2}r^{2}-a^{2}\right)d\varphi^{2},$$

(52)

always the cylindrical coordinates $(t,r,\varphi)$ are with usual ranges, and $\alpha=1-4\mu$ is the wedge parameter with $\alpha<1.$ Here, the rotation parameter $a=4J$ has units of distance with $J$ stands for the angular momentum of the string.

The covariant and contravariant components of the metric tensor are

$$g_{\mu\nu}=\left(\begin{array}[]{ccc}1&0&a\\ 0&-1&0\\ a&0&-\left(\alpha^{2}r^{2}-a^{2}\right)\end{array}\right),\quad g^{\mu\nu}=\left(\begin{array}[]{ccc}1-\frac{a^{2}}{\alpha^{2}r^{2}}&0&\frac{a}{\alpha^{2}r^{2}}\\ 0&-1&0\\ \frac{a}{\alpha^{2}r^{2}}&0&-\frac{1}{\alpha^{2}r^{2}}\end{array}\right).$$

(53)

To study the relativistic quantum motion of a scalar boson interacting with the gravitational field of the background geometry defined by the metric 52, we need to write the generic KG equation in the curved space in question, given by

$$\left(\square+m^{2}-\xi R\right)\Psi(t,r,\varphi)=0,$$

(54)

where $\Psi$ is a two-components vector

$$\Psi(x)=\left(\begin{array}[]{c}\phi_{1}\left(x\right)\\ \phi_{2}\left(x\right)\end{array}\right),\qquad x\equiv(t,r,\varphi).$$

(55)

satisfying the condition

$$\psi=\phi_{1}+\phi_{2},\qquad i\mathcal{D}^{\prime}\psi=\mathcal{N}\left(\phi_{1}-\phi_{2}\right),$$

(56)

with $\mathcal{D}^{\prime}=\frac{\partial}{\partial t}+\mathcal{Y^{\prime}}.$

The next step is to develop the approach used in Sec III for the case of rotating cosmic strings. We adopt the GFVT to derive the equations of motion for this problem, then we solve them to obtain the wave functions and the energy spectra. Following Ref. (key-52, ; key-53, ) , the identification of the Hamiltonian leads to the idea of employing a two-components formulation of the KG type fields. In order to rewrite Eq. (54) in the Hamiltonian form, one requires to introduce new definitions for the quantities presented in Eqs. ((29)), ((30)) and ((28)). In the case under consideration, we have

$$\mathcal{H}_{GFVT}^{\prime}=\tau_{z}\left(\frac{\mathcal{N}^{2}+\mathcal{T^{\prime}}}{2\mathcal{N}}\right)+i\tau_{y}\left(\frac{-\mathcal{N}^{2}+\mathcal{T^{\prime}}}{2\mathcal{N}}\right)-i\mathcal{Y}^{\prime},$$

(57)

	$\displaystyle\mathcal{T}^{\prime}$	$\displaystyle=\partial_{i}\frac{G^{ij}}{g^{00}}\partial_{j}+\frac{m^{2}-\xi R}{g^{00}}+\frac{1}{\mathcal{F}}\nabla_{i}\left(\sqrt{-g}G^{ij}\right)\nabla_{j}\left(\frac{1}{\mathcal{F}}\right)+\sqrt{\frac{\sqrt{-g}}{g^{00}}}G^{ij}\nabla_{i}\nabla_{j}\left(\frac{1}{\mathcal{F}}\right)+\frac{1}{4\mathcal{F}^{4}}\left[\nabla_{i}\left(\mathcal{U}^{i}\right)\right]^{2}$
		$\displaystyle\qquad-\frac{1}{2\mathcal{F}^{2}}\nabla_{i}\left(\frac{g^{0i}}{g^{00}}\right)\nabla_{j}\left(\mathcal{U}^{i}\right)-\frac{g^{0i}}{2g^{00}\mathcal{F}^{2}}\nabla_{i}\nabla_{j}\left(\mathcal{U}^{i}\right),$		(58)

$$\mathcal{Y}^{\prime}=\frac{1}{2}\left\{\partial_{i},\sqrt{-g}\frac{g^{0i}}{g^{00}}\right\},$$

(59)

where

$$G^{ij}=g^{ij}-\frac{g^{0i}g^{0j}}{g^{00}},\qquad\mathcal{F}=\sqrt{g^{00}\sqrt{-g}},\qquad\mathcal{U}^{i}=\sqrt{-g}g^{0i}.$$

(60)

It has been shown in Ref. (key-53, ) that the transformations ((57)), ((58)) and (59) with the definitions ((60)) are exact and covers any inertial and gravitational fields. It should be noted that these exact transformations ensures to obtain the block-diagonal form of the Hamiltonian $\mathcal{H}_{GFVT}$ which does not depend on the parameter $\mathcal{N}.$ For the geometry ((53)), the eigenvalues of the operator $\mathcal{Y}^{\prime}$ are given by

$$\mathcal{Y}^{\prime}\Psi\left(t,r,\varphi\right)=\frac{1}{2}\left\{\partial_{2},\sqrt{-g}\frac{g^{02}}{g^{00}}\right\}\Psi\left(t,r,\varphi\right)=\frac{g^{02}}{g^{00}}\frac{\partial}{\partial\varphi}\Psi\left(t,r,\varphi\right),$$

(61)

where the field $\Psi$ obeys the non-unitary transformation $\Psi\equiv\Phi^{\prime}=\mathcal{F}\Phi$ which permits to obtain pseudo-Hermitian Hamiltonian $\mathcal{H}_{GFVT}^{\prime}=\mathcal{F}\mathcal{H}_{GFVT}^{\prime}\mathcal{F}^{-1},$ $\mathcal{H}_{GFVT}^{\prime}=\tau_{z}\left(\mathcal{H}_{GFVT}^{\prime}\right)^{\dagger}\tau_{z}.$

After some mathematical calculations, Eq. ((58)) takes the form

$$\mathcal{T}^{\prime}=-\frac{1}{\mathcal{F}}\left[\left(\frac{\partial}{\partial r}\right)\left(\sqrt{-g}\frac{\partial}{\partial r}\right)+\frac{\sqrt{-g}}{\alpha^{2}r^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}\right]\frac{1}{\mathcal{F}}+\frac{m^{2}}{g^{00}}-\left(\frac{g^{02}}{g^{00}}\frac{\partial}{\partial\varphi}\right)^{2},$$

(62)

where for the metric ((52)) the Ricci scalar vanishes i.e, $R=0$.

Using the two components fields (55) and the Hamiltonian ((57)), we can express the field equation ((54)) as the Schrodinger’s equation

$$\mathcal{H}_{GFVT}^{\prime}\Psi^{\prime}(x)=i\,\frac{d}{dt}\Psi^{\prime}(x),\qquad x\equiv(t,r,\varphi).$$

(63)

To solve this eigenvalue problem, let us consider for the wave function the Ansatz below

$$\Psi(x)=\mathcal{F}\Phi\left(x\right)=\mathcal{F}\left(\begin{array}[]{c}\phi_{1}\left(r\right)\\ \phi_{2}\left(r\right)\end{array}\right)e^{-\left(iEt-j\varphi\right)},$$

(64)

substituting ((64)) into Eq. ((63)), we find the following coupled differential equations

	$\displaystyle\mathcal{N}^{2}\mathcal{F}\left(\phi_{1}-\phi_{2}\right)+\mathcal{T}^{\prime}\mathcal{F}\left(\phi_{1}-\phi_{2}\right)+2\mathcal{N}\left(\frac{g^{02}}{g^{00}}j\right)\mathcal{F}\phi_{1}$	$\displaystyle=2\mathcal{N}E\mathcal{F}\phi_{1}$		(65)
	$\displaystyle\mathcal{N}^{2}\mathcal{F}\left(\phi_{1}-\phi_{2}\right)-\mathcal{T}^{\prime}\mathcal{F}\left(\phi_{1}+\phi_{2}\right)+2\mathcal{N}\left(\frac{g^{02}}{g^{00}}j\right)\mathcal{F}\phi_{2}$	$\displaystyle=2\mathcal{N}E\mathcal{F}\phi_{2},$

We can add and subtract the two equations of ((65)) to get, respectively,

$$\begin{aligned} \mathcal{N}\mathcal{F}\left(\phi_{1}-\phi_{2}\right)+\left(\frac{g^{02}}{g^{00}}j\right)\mathcal{F}\left(\phi_{1}+\phi_{2}\right)&=E\mathcal{F}\left(\phi_{1}+\phi_{2}\right)\end{aligned},$$

(66)

and

$$-\mathcal{T}^{\prime}\mathcal{F}\left(\phi_{1}+\phi_{2}\right)+\mathcal{N}\left(\frac{g^{02}}{g^{00}}j\right)\mathcal{F}\left(\phi_{1}-\phi_{2}\right)=\mathcal{N}E\mathcal{F}\left(\phi_{1}-\phi_{2}\right),$$

(67)

where we have used the relation

$$\phi_{1}=\frac{1}{\mathcal{N}}\left(E-\frac{g^{02}}{g^{00}}j\right)\phi_{2},$$

(68)

After simple algebraic manipulations we arrive at the following second order differential equation for the radial function $\psi(r)$

$$\left[\frac{1}{\sqrt{-g}}\frac{\partial}{\partial r}\left(\sqrt{-g}\frac{\partial}{\partial r}\right)-\frac{\left(aE+j\right)^{2}}{\alpha^{2}r^{2}}+E^{2}-m^{2}\right]\psi\left(r\right)=0,$$

(69)

setting $\varsigma^{2}=\frac{\left(aE+j\right)^{2}}{\alpha^{2}},\gamma=E^{2}-m^{2},$ yields

$$\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{\varsigma^{2}}{r^{2}}+\gamma\right]\varphi_{s}\left(r\right)=0,$$

(70)

Eq. ((70)) is expressed as a Bessel differential equation, its solutions can be written in terms of Bessel function of first kind as

$$\psi\left(r\right)=A^{\prime}\,J_{\frac{aE+j}{\alpha}}\left(\sqrt{E^{2}-m^{2}}\,r\right),$$

(71)

where $A^{\prime}$ is an integration constant.

The complete eigenstates are given by

$$\psi\left(t,r,\varphi\right)=|C^{\prime}|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\ 1-\frac{E}{\mathcal{N}}\end{array}\right)e^{-i\left(Et-j\varphi\right)}\,J_{\frac{aE+j}{\alpha}}\left(\sqrt{E^{2}-m^{2}}\,r\right),$$

(72)

From now on we proceed to study the Klein Gordon oscillator in a rotating cosmic string space-time . Firstly, we start by considering a scalar quantum particle embedded in the background gravitational field of the space-time described by the metric ((52)). In this way, we shall introduce a replacement of the momentum operator $p_{i}\longrightarrow p_{i}+im\omega x_{i}$ where $p_{i}=i\nabla_{i}$ in Eq. ((58)). Then, we have

$$\mathcal{T}^{\prime}=-\frac{1}{\mathcal{F}}\left[\left(\frac{\partial}{\partial r}-m\omega r\right)\left(\sqrt{-g}\left(\frac{\partial}{\partial r}+m\omega r\right)\right)+\frac{\sqrt{-g}}{\alpha^{2}r^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}\right]\frac{1}{\mathcal{F}}+\frac{m^{2}}{g^{00}}-\left(\frac{g^{02}}{g^{00}}\frac{\partial}{\partial\varphi}\right)^{2},$$

(73)

Based on the previous analyses, we shall apply the GFVT for the case of KGO in the concerned space through the same steps used before. Inserting Eq. ((73)) and Eq. ((61)) into the Hamiltonian ((57)), then assuming the solution ((64)) gives two coupled differential equations similar to those of Eq. ((65)) but with different value of $\mathcal{T}^{\prime}$.

Manipulating exactly the same steps before, we obtain the following radial equation

$$\left[\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}-m^{2}\omega^{2}r^{2}-\frac{\vartheta^{2}}{r^{2}}+\delta\right]\psi\left(r\right)=0,$$

(74)

where we have defined

$$\vartheta^{2}=\left(\frac{aE+j}{\alpha}\right)^{2},\qquad\delta=E^{2}-m^{2}+2m\omega.$$

(75)

To solve Eq. ((74)), we introduce a new dimensionless variable $\mathcal{R}=m\omega r^{2}$, and by substitution into Eq. (74) , the resulting equation reads

$$\left[\frac{d^{2}}{d\mathcal{R}^{2}}+\frac{1}{\mathcal{R}}\frac{d}{d\mathcal{R}}-\frac{\vartheta^{2}}{4\mathcal{R}^{2}}-\frac{1}{4}+\frac{\delta}{4m\omega\mathcal{R}}\right]\psi\left(\mathcal{R}\right)=0,$$

(76)

To eliminate the term $d\psi(\mathcal{R})\big{/}d\mathcal{R},$ we consider the following change of variable

$$\psi(\mathcal{R})=\mathcal{R}^{-\frac{1}{2}}\Xi\left(\mathcal{R}\right),$$

(77)

Eq. ((76)) then becomes

$$\frac{d^{2}\Xi\left(\mathcal{R}\right)}{d\mathcal{R}^{2}}+\left[-\frac{1}{4}+\frac{\delta}{4m\omega\mathcal{R}}+\frac{\frac{1}{4}-\left(\frac{\vartheta}{2}\right)^{2}}{\mathcal{R}^{2}}\right]\Xi\left(\mathcal{R}\right)=0,$$

(78)

and it has the form of the Whittaker differential equation (key-55, ). The general solution of this equation, which is regular at the origin, is given by

$$\Xi\left(\mathcal{R}\right)=|C|\mathcal{R}^{-\tfrac{1}{2}}M\left(\frac{\delta}{4m\omega},\frac{|\vartheta|}{2},\mathcal{R}\right),$$

(79)

where $|C|$ is an arbitrary constant and $M\left(\frac{\delta}{4m\omega},\frac{|\vartheta|}{2},\mathcal{R}\right)$ is the Whittaker M-function defined via the confluent hyper-geometric functions as

$$M\left(\frac{\delta}{4m\omega},\frac{|\vartheta|}{2},\mathcal{R}\right)=e^{-\frac{\mathcal{R}}{2}}\mathcal{R}^{\frac{\left|\vartheta\right|+1}{2}}{}_{1}F_{1}\left(\frac{|\vartheta|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2},|\vartheta|+1,\mathcal{R}\right),$$

(80)

Using the definition ((80)), the final expression of the wave-function of the spinless KGO propagating in the rotating cosmic string background can be represented as

$$\psi\left(t,r,\varphi\right)=|\mathtt{C}|\left(\begin{array}[]{c}1+\frac{E}{\mathcal{N}}\\ 1-\frac{E}{\mathcal{N}}\end{array}\right)\left(m\omega r^{2}\right)^{\frac{\left|\vartheta\right|}{2}}e^{-\frac{m\omega}{2}r^{2}}e^{-i\left(Et-j\varphi\right)}{}_{1}F_{1}\left(\frac{|\vartheta|}{2}-\frac{\delta}{4m\omega}+\frac{1}{2},|\vartheta|+1,m\omega r^{2}\right),$$

(81)

where the parameters $\vartheta$ and $\delta$ are defined in Eq. ((75)).